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Multi-Fractal Formalism for Quasi-Self-Similar Functions

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Abstract

The study of multi-fractal functions has proved important in several domains of physics. Some physical phenomena such as fully developed turbulence or diffusion limited aggregates seem to exhibit some sort of self-similarity. The validity of the multi-fractal formalism has been proved to be valid for self-similar functions. But, multi-fractals encountered in physics or image processing are not exactly self-similar. For this reason, we extend the validity of the multi-fractal formalism for a class of some non-self-similar functions. Our functions are written as the superposition of “similar” structures at different scales, reminiscent of some possible modelization of turbulence or cascade models. Their expressions look also like wavelet decompositions. For the computation of their spectrum of singularities, it is unknown how to construct Gibbs measures. However, it suffices to use measures constructed according the Frostman's method. Besides, we compute the box dimension of the graphs.

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Authors and Affiliations

  1. Département de Mathématiques, Faculté des Sciences de Bizerte, 7021, Jarzouna, Bizerte, Tunisia

    Jamil Aouidi

  2. Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire, 1060, Tunis, Tunisia

    Mourad Ben Slimane

Authors
  1. Jamil Aouidi
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  2. Mourad Ben Slimane
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Aouidi, J., Ben Slimane, M. Multi-Fractal Formalism for Quasi-Self-Similar Functions. Journal of Statistical Physics 108, 541–590 (2002). https://doi.org/10.1023/A:1015729908167

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